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Question #43324

Difficulty: 8/10 | 🟢 Free

The Question

An ideal gas with adiabatic exponent $\gamma$ (dimensionless) is confined in a horizontal cylinder of cross-sectional area $A \text{ (m}^2\text{)}$ . The gas is initially at equilibrium with volume $V_0 \text{ (m}^3\text{)}$ and pressure $P_0 \text{ (Pa)}$ . A frictionless piston of mass $m \text{ (kg)}$ seals the gas. An external driving force $F(t) = F_0 \cos(\omega t) \text{ (N)}$ is applied to the piston, where $\omega \text{ (rad/s)}$ is the angular frequency. Assuming the process is strictly adiabatic and displacements are small, what is the root-mean-square (RMS) value of the fluctuating component of the gas pressure, $\Delta P_{rms} \text{ (Pa)}$ , in the steady state?

Options

A.
$\frac{\gamma P_0 A F_0}{\sqrt{2} |\gamma P_0 A^2 - m \omega^2 V_0|}$
B.
$\frac{F_0}{\sqrt{2} A}$
C.
$\frac{P_0 A F_0}{\sqrt{2} |P_0 A^2 - m \omega^2 V_0|}$
D.
$\frac{\gamma P_0 A F_0}{\sqrt{2} (\gamma P_0 A^2 + m \omega^2 V_0)}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Instantaneous vs RMS vs Peak|Adiabatic Process Constraint|State Function vs Path Function

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple thermodynamic constraints with mechanical oscillations.

A common trap is assuming the process is quasi-static or ignoring the piston's inertia, leading students to incorrectly equate the RMS gas pressure fluctuation directly to the RMS external force per unit area ( $\Delta P_{rms} = F_{rms}/A$ ).

By forcing the use of the adiabatic bulk modulus and Newton's second law, the problem reveals that the restorative force is frequency-dependent.

The correct approach requires deriving the steady-state mechanical displacement amplitude first, and then mapping it back to the thermodynamic pressure variation via the adiabatic constraint $\Delta P = -\gamma P \Delta V / V$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Instantaneous vs RMS vs Peak
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free